Regularization which uses the derivatives at a cross of four points around each pixel centre and values adapted to the data being fitted to smooth an inversion’s solution.

An adaptive regularization scheme which splits every source pixel into a cross of four regularization points and interpolates to these points in order to smooth an inversion’s solution.

The size of this cross is determined via the size of the source-pixel, for example if the source pixel is a Voronoi pixel the area of the pixel is computed and the distance of each point of the cross is given by the area times 0.5.

For the weighted regularization scheme, each pixel is given an ‘effective regularization weight’, which is applied when each set of pixel neighbors are regularized with one another. The motivation of this is that different regions of a pixelization’s mesh require different levels of regularization (e.g., high smoothing where the no signal is present and less smoothing where it is, see (Nightingale, Dye and Massey 2018)).

Unlike `Constant` regularization, neighboring pixels must now be regularized with one another in both directions (e.g. if pixel 0 regularizes pixel 1, pixel 1 must also regularize pixel 0). For example:

B = [-1, 1] [0->1]

[-1, -1] 1 now also regularizes 0

For `Constant` regularization this would NOT produce a positive-definite matrix. However, for the weighted scheme, it does!

The regularize weight_list change the B matrix as shown below - we simply multiply each pixel’s effective regularization weight by each row of B it has a -1 in, so:

regularization_weights = [1, 2, 3, 4]

B = [-1, 1, 0 ,0] # [0->1]

[0, -2, 2 ,0] # [1->2] [0, 0, -3 ,3] # [2->3] [4, 0, 0 ,-4] # [3->0]

If our -1’s werent down the diagonal this would look like:

B = [4, 0, 0 ,-4] # [3->0]

[0, -2, 2 ,0] # [1->2] [-1, 1, 0 ,0] # [0->1] [0, 0, -3 ,3] # [2->3] This is valid!

A full description of regularization and this matrix can be found in the parent AbstractRegularization class.

Parameters:
• coefficients – The regularization coefficients which controls the degree of smoothing of the inversion reconstruction in high and low signal regions of the reconstruction.

• signal_scale (`float`) – A factor which controls how rapidly the smoothness of regularization varies from high signal regions to low signal regions.

Methods

 `regularization_matrix_from` Returns the regularization matrix with shape [pixels, pixels]. `regularization_weights_from` Returns the regularization weights of this regularization scheme.
regularization_matrix_from(linear_obj)[source]#

Returns the regularization matrix with shape [pixels, pixels].

Parameters:

linear_obj (`LinearObj`) – The linear object (e.g. a `Mapper`) which uses this matrix to perform regularization.

Return type:

The regularization matrix.